Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $\alpha$, the arithmetic mean of the first $n$ continued fraction digits $a_i(\alpha)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $\alpha$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $\alpha$, providing rigorous results, as well as numerically supported conjectures.Comment: 32 pages, 7 figures. Substantial additions were made to previous version, including Theorem 1.3, Section 6, and Appendix

Partial nonlinear reciprocity breaking through ultrafast dynamics in a random photonic medium

We demonstrate that ultrafast nonlinear dynamics gives rise to reciprocity breaking in a random photonic medium. Reciprocity breaking is observed via the suppression of coherent backscattering, a manifestation of weak localization of light. The effect is observed in a pump-probe configuration where the pump induces an ultrafast step-change of the refractive index during the dwell time of the probe light in the material. The dynamical suppression of coherent backscattering is reproduced well by a multiple scattering Monte Carlo simulation. Ultrafast reciprocity breaking provides a distinct mechanism in nonlinear optical media which opens up avenues for the active manipulation of mesoscopic transport, random lasers, and photon localization.Comment: 5 pages, 4 figure

Coherent backscattering of Bose-Einstein condensates in two-dimensional disorder potentials

We study quantum transport of an interacting Bose-Einstein condensate in a two-dimensional disorder potential. In the limit of vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. We reproduce this result, obtained from the numerical integration of the Gross-Pitaevskii equation, by a diagrammatic theory of weak localization in presence of a nonlinearity.Comment: 4 pages, 4 figure

Transient entrainment and interruption of atrioventricular node tachycardia

The possibility of transiently entraining and interrupting the common type of atrioventricular (AV) node tachycardia (anterograde slow, retrograde fast AV node pathway) was studied using atrial and ventricular pacing in 18 patients with paroxysmal AV node tachycardia. Transient entrainment occurred in all patients. During atrial pacing, localized block in the AV node for one beat followed by anterograde conduction over the fast pathway was observed in three patients. During ventricular pacing, localized block for one beat followed by retrograde conduction over the slow pathway was not observed in any patient. Neither atrial nor ventricular fusion beats were observed during entrainment.These observations indicate in a way not previously shown that reentry involving two functionally dissociated pathways in the AV node is the underlying mechanism of paroxysmal AV node tachycardia. The inability to demonstrate atrial or ventricular fusion beats during entrainment suggests a true intranodal location of the reentrant circuit. Finally, the ability to transiently entrain intranodal tachycardia demonstrates that this electrophysiologic phenomenon is not exclusively limited to macroreentrant circuits

Emission of photon echoes in a strongly scattering medium

We observe the two- and three-pulse photon echo emission from a scattering powder, obtained by grinding a Pr$^{3+}$:Y$_2$SiO$_5$ rare earth doped single crystal. We show that the collective emission is coherently constructed over several grains. A well defined atomic coherence can therefore be created between randomly placed particles. Observation of photon echo on powders as opposed to bulk materials opens the way to faster material development. More generally, time-domain resonant four-wave mixing offers an attractive approach to investigate coherent propagation in scattering media

Coherent propagation of waves in random media with weak nonlinearity

We develop a diagrammatic theory for transport of waves in disordered media with weak nonlinearity. We first represent the solution of the nonlinear wave equation as a nonlinear Born series. From this, we construct nonlinear ladder and crossed diagrams for the average wave intensity. Then, we sum up the diagrammatic series completely, i.e. nonperturbatively in the strength of the nonlinearity, and thereby obtain integral equations describing both nonlinear diffusive transport and coherent backscattering of the average intensity. As main result, we find that the nonlinearity significantly influences the magnitude of the coherent backscattering effect. Depending on the type of nonlinearity, coherent backscattering is either enhanced or suppressed, as compared to the linear case.Comment: 20 pages, 12 figure

Binegativity and geometry of entangled states in two qubits

We prove that the binegativity is always positive for any two-qubit state. As a result, as suggested by the previous works, the asymptotic relative entropy of entanglement in two qubits does not exceed the Rains bound, and the PPT-entanglement cost for any two-qubit state is determined to be the logarithmic negativity of the state. Further, the proof reveals some geometrical characteristics of the entangled states, and shows that the partial transposition can give another separable approximation of the entangled state in two qubits.Comment: 5 pages, 3 figures. I made the proof more transparen

Evaluating the implementation fidelity to a successful nurse-led model (INTERCARE) which reduced nursing home unplanned hospitalisations

Implementation fidelity assesses the degree to which an intervention is delivered as it should be. Fidelity helps to determine if the outcome(s) of an intervention are attributed to the intervention itself or to a failure of its implementation. Little is known about how fidelity impacts the intended outcome(s) and what elements or moderators can affect the fidelity trajectory over time. We exemplify the meaning of implementation fidelity with INTERCARE, a nurse-led care model that was implemented in eleven Swiss nursing homes (NHs) and showed effectiveness in reducing unplanned hospital transfers. INTERCARE comprises six core elements, including advance care planning and tools to support inter- and interprofessional communication, which were introduced with carefully developed implementation strategies.; A mixed-methods convergent/triangulation design was used to investigate the influence of implementation fidelity on unplanned transfers. A fidelity questionnaire measuring the degree of fidelity to INTERCARE's core components was fielded at four time points in the participating NHs. Two-monthly meetings were conducted with NHs (September 2018-January 2020) and structured notes were used to determine moderators affecting fidelity (e.g., participant responsiveness). We used the fidelity scores and generalized linear mixed models to analyze the quantitative data. The Framework method was used for the qualitative analysis. The quantitative and qualitative findings were integrated using triangulation.; A higher overall fidelity score showed a decreasing rate of unplanned hospital transfers post-intervention (OR: 0.65 (CI = 0.43-0.99), p = 0.047). A higher fidelity score to advance care planning was associated with lower unplanned transfers (OR = 0.24 (CI 0.13-0.44), p = < 0.001) and a lower fidelity score for communication tools (e.g., ISBAR) to higher rates in unplanned transfers (OR = 1.69 (CI 1.30-2.19), p = < 0.003). In-house physicians with a collaborative approach and staff's perceived need for nurses working in extended roles, were important moderators to achieve and sustain high fidelity.; Implementation fidelity is challenging to measure and report, especially in complex interventions, yet is crucial to better understand how such interventions may be tailored for scale-up. This study provides both a detailed description of how fidelity can be measured and which ingredients highly contributed to reducing unplanned NH transfers.; The INTERCARE study was registered at clinicaltrials.gov Protocol Record NCT03590470